550 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL Global Stability of Relay Feedback Systems

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1 550 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 Global Stability of Relay Feedback Systems Jorge M. Gonçalves, Alexre Megretski, Munther A. Dahleh Abstract For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. This paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincaré maps associated with RFS. These results are based on the discovery that a typical Poincaré map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. Index Terms Global asymptotic stability, limit cycles, piecewise linear systems, Poincaré maps, relay feedback systems. I. INTRODUCTION IT IS OFTEN possible to linearize a system, i.e., to obtain a linear representation of its behavior. That representation approximates the true dynamics well in a small region. For example, the true equations of the pendulum are never linear but, for very small deviations (a few degrees) they may be satisfactorily replaced by linear equations. In other words, for small deviations, the pendulum may be replaced by a harmonic oscillator. This ceases to hold, however, for large deviations, in dealing with these, one must consider the nonlinear equation itself not merely a linear substitute. In this work we are interested in Manuscript received October 26, 1999; revised October 11, Recommended by Associate Editor B. Bernhardsson. The work of J. M. Gonçalves was supported in part by the Portuguese Fundação para a Ciência e Tecnologia under the program PRAXIS XXI. The work of J. M. Gonçalves A. Megretski was supported in part by the National Science Foundation (NSF) under Grants ECS , ECS , ECS , in part by the Air Force Office of Scientific Research (AFOSR) under Grants F F The work of J. M. Gonçalves M. A. Dahleh was supported in part by the NSF under Grant ECS in part by the AFOSR under Grants AFOSR F F The authors are with the Department of Electrical Engineering Computer Science, Massachusetts Institute of Technology, Cambridge, MA ( jmg@mit.edu; ameg@mit.edu; dahleh@lids.mit.edu). Publisher Item Identifier S (01) a class of nonlinear systems known as piecewise linear systems (PLS). PLS are characterized by a finite number of linear dynamical models together with a set of rules for switching among these models. Therefore, this model description causes a partitioning of the state space into cells. These cells have distinctive properties in that the dynamics within each cell are described by linear dynamic equations. The boundaries of each cell are in effect switches between different linear systems. Those switches arise from the breakpoints in the piecewise linear functions of the model. The reason why we are interested in studying this class of systems is to capture discontinuity actions in the dynamics from either the controller or system nonlinearities. On one h, a wide variety of physical systems are naturally modeled this way due to real-time changes in the plant dynamics like collisions, friction, saturation, walking robots, etc. On the other h, an engineer can introduce intentional nonlinearities to improve system performance, to effect economy in component selection, or to simplify the dynamic equations of the system by working with sets of simpler equations (e.g., linear) switch among these simpler models (in order to avoid dealing directly with a set of nonlinear equations). Examples include control of inverted pendulums [3], control of antilock brake systems [21], control of missile autopilots [7], control of autopilot of aircrafts [23], auto-tuning of PID regulators using relays [4], etc. Although widely used, very few results are available to analyze most PLS. More precisely, one typically cannot guarantee stability, robustness, performance properties of PLS designs. Rather, any such properties are inferred from extensive computer simulations. However, in the absence of rigorous analysis tools, PLS designs come with no guarantees. In other words, complete systematic analysis design methodologies have yet to emerge. This paper introduces a new methodology to globally analyze PLS using quadratic surface Lyapunov functions. This methodology is based in finding quadratic Lyapunov functions on associated switching surfaces that can be used to prove that a map from one switching surface to the next switching surface is contracting in some norm. The novelty of this work is based on expressing maps induced by an LTI flow between two switching surfaces as linear transformations analytically parametrized by a scalar function of the state. Furthermore, level sets of this function are convex subsets of linear manifolds with dimension lower than the one of the switching surfaces. The search for global quadratic Lyapunov functions on switching surfaces is then done by solving a set of LMIs, which can be efficiently done using available computational tools. The main difference between this previous work [17], [20], [15], is that we look for quadratic Lyapunov functions on switching surfaces instead of quadratic Lyapunov functions in /01$ IEEE

2 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 551 the state space. An immediate advantage is that this allows us to analyze not only equilibrium points (recently, we proved global asymptotic stability of on/off systems [10] saturation systems [11]) but also limit cycles. Another advantage is that, for a given PLS, complexity of analysis is the same for high-dimension or low-dimension systems. In [17], [20], [15], partitioning of the state-space is the key in this approach. For most PLS, construction of piecewise quadratic Lyapunov functions is only possible after a more refined partition of the state space, in addition to the already existent natural state-space partition of the PLS. As a consequence, the analysis method is efficient only when the number of partitions required to prove stability is small. As illustrated in an example in [9], however, even for second order systems, the method can become computationally intractable. Also, for high-order systems, it is extremely hard to obtain a refinement of partitions in the state-space to efficiently analyze PLS. To demonstrate the success of this methodology, we apply it to a simple yet very hard to analyze class of PLS known as relay feedback systems (RFS). Although the focus of this paper is on RFS, it is important to point out that most ideas behind the main results described here can be used in the analysis of more general PLS. Analysis of RFS is a classic field. The early work was motivated by relays in electromechanical systems simple models of dry friction. Applications of relay feedback range from stationary control of industrial processes to control of mobile objects as used, for example, in space research. A vast collection of applications of relay feedback can be found in the first chapter of [24]. More recent examples include the delta sigma modulator (as an alternative to conventional A/D converters) the automatic tuning of PID regulators. In the delta sigma modulator, a relay produces a bit stream output whose pulse density depends on the applied input signal amplitude (see, for example, [1]). Various methods were applied to the analysis of delta sigma modulators. In most situations, however, none allowed to verify global stability of nonlinear oscillations. As for the automatic tuning of PID regulators, implemented in many industrial controllers, the idea is to determine some points on the Nyquist curve of a stable open loop plant by measuring the frequency of oscillation induced by a relay feedback (see, for example, [4]). One problem that needs to be solved here is the characterization of those systems that have unique global attractive unimodal limit cycles. This problem is important because it gives the class of systems where relay tuning can be used. Some important questions can be asked about RFS. Do they have limit cycles? If so, are they locally stable or unstable? If there exists a unique locally stable limit cycle, is it also globally stable? Over many years, researchers have been trying to answer these questions. References [5], [24], [19] survey a number of analysis methods. Rigorous results on existence local stability of limit cycles of RFS can be found in [2], [16], [25], [8]. Reference [2] presents necessary sufficient conditions for local stability of limit cycles. Reference [16] emphasizes fast switches their properties also proves volume contraction of RFS. In [12], reasonably large regions of stability around limit cycles were characterized. For second-order systems, convergence analysis can be done in the phase-plane [22], [14]. Stable second-order nonminimum phase processes can in this way be shown to have a globally attractive limit cycle. In [18], it is proved that this also holds for processes having an impulse response sufficiently close, in a certain sense, to a second-order nonminimum phase process. Many important RFS, however, are not covered by this result. It is then clear that the problem of rigorous global analysis of relay-induced oscillations is still open. In this paper, we prove global stability of symmetric unimodal 1 limit cycles of RFS by finding quadratic surface Lyapunov functions for associated Poincaré maps. These results are based on the discovery that typical Poincaré maps associated with RFS can be represented as linear transformations parametrized by a scalar function of the state. Quadratic stability can then be easily checked by solving a set of LMIs, which can be efficiently done using available computational tools. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. Note that although the stability analysis in this paper focuses on symmetric unimodal limit cycles, similar ideas can be applied to prove stability of other types of limit cycles. As we will see, analysis of symmetric unimodal limit cycles can be done by analyzing a single map from one switching surface to the other switching surface. Other types of limit cycles require a simultaneous analysis of several maps from one switching surface to the other switching surface. Multiple maps, however, have been shown in [10] [11] to work as well as the single map described in this paper. This paper is organized as follows. Section II starts by giving some mathematical preliminaries, including definitions of some stard concepts. Section III gives some background on RFS followed by the main results of this paper (Section IV). There, we first show that Poincaré maps can be represented as linear transformations, then use this result to demonstrate that quadratic stability of Poincaré maps can be easily checked by solving sets of LMIs. Section V contains some illustrative examples. Improvements of the stability condition presented in Section IV are discussed in Section VI. Section VII considers several computationally issues associated with the stability results,, finally, conclusions future work are discussed in Section VIII. II. MATHEMATICAL PRELIMINARIES The purpose of this section is to briefly introduce several mathematical concepts tools that will be used throughout the paper. Mathematical tools like linear matrix inequalities a simple version of the -procedure are the engines behind the 1 Symmetric unimodal limit cycles are those that are symmetric about the origin switch only twice per cycle.

3 552 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 stability results presented later in the paper. For this reason, these topics are briefly introduced for completeness. A. Stard Notation Let the field of real numbers be denoted by, the set of vectors with elements in by, the set of all matrices with elements in by. Let denote the identity matrix superscript denote transpose. A matrix is called symmetric if positive definite (positive semidefinite) if ( ) for all nonzero. on sts for for all nonzero. A matrix is Hurwitz if the real part of each eigenvalue of is negative. The 2-norm of is given by. Let denote the space of all real-valued functions on such that. A set is convex if whenever, is a cone if implies for any. A function is piecewise constant if there exists a sequence of points with as, as, such that the function is constant in. Let st for the for the. B. Linear Matrix Inequalities the -Procedure An LMI has the form where is the variable the symmetric matrices, are given. The LMI (1) is a convex constraint on, i.e., the set is convex. Expressing solutions to problems in terms of LMIs is a common practice these days. Mathematical software tools capable of efficiently finding satisfying (1) are available. The strategy in this paper is to express the problem of global analysis of relay-induced oscillations as LMIs. One tool that will be useful later in the paper is the -procedure. Here, we describe a simple version of this tool. Let be quadratic forms of the variable, where. Assume there exists an such that. Then, the following condition on for all holds if only if there exists a such that such that for all. For more information on LMIs the -procedure the reader is referred, for example, to [6]. III. BACKGROUND In this section, we start by defining RFS talking about some of their properties. Then, we present some relevant results from the literature on existence local stability of limit cycles of RFS. Finally, we define Poincaré maps for RFS. (1) Fig. 1. A. Definitions Relay feedback system. Consider a single-input single-output (SISO) LTI system satisfying the following linear dynamic equations: where is a Hurwitz matrix, in feedback with a relay (see Fig. 1) where is the hysteresis parameter. By a solution of (2) (3) we mean functions satisfying (2) (3), where is piecewise constant if if if or, or, or is a switching time of a solution of (2) (3) if is discontinuous at. We say a trajectory of (2) (3) switches at some time if is a switching time. In the state space, the switching surfaces of the RFS are the surfaces of dimension where is equal to, respectively. More precisely Consider a subset of given by This set is important since it characterizes those points in that can be reached by any trajectory starting at. We call it the departure set in (see Fig. 2). Similarly, define as This is the arrival set in. It is easy to see that, where sts for the set. B. Existence of Solutions If an initial condition does not belong to a switching surface then existence of solution is guaranteed at least from the initial condition to the first intersection with a switching surface. This (2) (3)

4 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 553 Fig. 2. The departure set. Fig. 4. Symmetry around the origin. Fig. 3. Existence of solutions when. follows since, in that region, the system is affine linear. When an initial condition belongs to a switching surface, however, depending on the RFS, a solution may or may not exist. If then existence of solution is always guaranteed since there is a gap between both switching surfaces. This gap allows a trajectory to evolve according to an affine system. In the case of the ideal relay, i.e., when, for some RFS there are initial conditions for which no solution exists. In Fig. 3, we have two examples of ideal RFS. The figure shows the vector field along both sides of the unique switching surface. Above, the vector field is given by, below by. are those points in such that, respectively. On the left in Fig. 3,, on the right. When, every point in has at least one solution. For an initial condition on the left of, the trajectory moves downwards, on the right of it moves upwards. In between, the trajectory can either move upwards or downwards. When, however, there is no solution if a trajectory starts between. The reason for this is that the vector field on both sides of the switching surface points toward the switching surface. In these situations, one of the following two alternatives is typically used to guarantee existence of solutions: 1) an hysteresis with is introduced to avoid chattering; or 2) the definition of relay in (3) is slightly modified to allow trajectories to evolve in the switching surface, leading to the so-called sliding modes. Here, we consider the first case. Although sliding modes are not studied in this paper, we expect that such systems can be analyzed using the same ideas described here. Hence, according to the definition of relay in (3), existence of solutions is guaranteed if, or if, where is the smallest number such that (see [16] for details). Note that trajectories of starting at any point will converge to the equilibrium point. When connected in feedback with a relay, one of the following two possible scenarios will occur for a certain trajectory starting at : this will either cross at some time, or it will never cross. The last situation is not interesting to us since it does not lead to limit cycle trajectories. One way to ensure a switch is to have, although this is not a necessary condition for the existence of limit cycles. However, if we are looking for globally stable limit cycles, it is in fact necessary to have. Otherwise, a trajectory starting at would not converge to the limit cycle. Throughout the paper, it is assumed. As mentioned before, for a large class of processes, there will be limit cycle oscillations. Let be a nontrivial periodic solution of (2) (3) with period, let be the limit cycle defined by the image set of. The limit cycle is called symmetric if. It is called unimodal if it only switches twice per cycle. A class of limit cycles of RFS we are particularly interested in is the class of symmetric unimodal limit cycles. The next proposition, proven in [2], gives necessary sufficient conditions for the existence of symmetric unimodal limit cycles. Proposition 3.1: Consider the RFS (2) (3). Assume there exists a symmetric unimodal limit cycle with period. Then the following conditions hold for Furthermore, the periodic solution condition given by C. Poincaré Maps of RFS (4) is obtained with the initial Before defining Poincaré maps, it is important to notice an interesting property of linear systems in relay feedback: their symmetry around the origin (see Fig.4). Proposition 3.2: Consider a trajectory of starting at. Then is a trajectory of starting at. Proof: Assume. Since is a trajectory of starting at.

5 554 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 Fig. 5. Definition of a Poincaré map for a RFS. This property tells us that, in terms of stability analysis, a limit cycle only needs to be studied from one switching surface (say ) to the other switching surface ( ). In other words, for analysis purposes, it is equivalent to consider the trajectory from to the next switch, or the trajectory starting at switching at. We then focus our attention on trajectories from to. Next, we define Poincaré maps for RFS. Typically, such maps are defined from one switching surface back to the same switching surface. In the case of RFS, however, a Poincaré map only needs to be defined as the map from one switching surface to the other switching surface, due to the symmetry of the system. Consider a symmetric unimodal limit cycle, with period, obtained with the initial condition. This means that a trajectory starting at crosses the switching surface at (see Fig. 5). To study the behavior of the system around the limit cycle we perturb by such that. Consider a solution of (2) (3) with initial condition let be its first switch. We are interested in studying the map from to (see Fig. 5). Note that this map is not continuous is multivalued. In general, there exist such that is not unique. This is illustrated in the next example. Example 3.1: Consider the RFS (2) (3) where the LTI system is given by Fig. 6. Existence of multiple solutions. Fig.7. is a -dimensional map. Let. Since then. Consider the multivalued Poincaré map defined by. Since is fixed, the Poincaré map can be redefined as the map given by, where. In result, is an equilibrium point of the discrete-time system (5) the hysteresis parameter is. Let,,,. The resulting can be seen in Fig. 6. When,. At this point, the trajectory can return to the region where (dash trajectory), or it can move into the region where with (dash dot trajectory). This means that a switch can occur at either or. Definition 3.1: Let. Define as the set of all times such that on. Define also the set of expected switching times as For instance, in the last example, initial condition. for the The following proposition, proven in [2], gives conditions for local stability of symmetric unimodal limit cycles. This result is based on the linearization of the Poincaré map around the origin. Proposition 3.3: Consider the RFS (2) (3). Assume there exists a symmetric unimodal limit cycle with period, obtained with the initial condition. Assume also the limit cycle is transversal 2 to at. The Jacobian of the Poincaré map at is given by where. The limit cycle is locally stable if has all its eigenvalues inside the unit disk. It is unstable if at least one of the eigenvalues of is outside the unit disk. In this paper, we are interested in systems that have a unique locally stable unimodal limit cycle. For such systems, the idea 2 is transversal to at if.

6 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 555 Fig. 8. Third-order nonminimum phase system. Fig. 9. Third-order minimum phase system. Fig. 10. Sixth-order system. is to construct a quadratic Lyapunov function on the switching surface to prove that the Poincaré map is globally stable. This, in turn, shows that the limit cycle is globally asymptotically stable. The next section shows that a Poincaré map from one switching surface to the other switching surface can be represented as a linear transformation analytically parametrized by the switching time. This representation will then allow us to reduce the problem of checking quadratic stability to the solution of a set of LMIs. IV. DECOMPOSITION AND STABILITY OF POINCARÉ MAPS This section contains the main results of this paper. Here, we show that a typical Poincaré map induced by an LTI flow between the switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This, in turn, allows us to reduce the problem of checking quadratic stability of Poincaré maps to the solution of a set of LMIs. Theorem 4.1: Consider the Poincaré map defined above. Let assume, for some all. Define for all (for, is defined by the limit as ). Then, for any there exists a such that Such is the switching time associated with. This theorem says that most Poincaré maps induced by an LTI flow between two hyperplanes can be represented as linear transformations analytically parametrized by a scalar function of the state. The advantage of expressing such maps this way is to have all nonlinearities depending only on one parameter. Although depends on, once is fixed, the map becomes linear in. Note that defined above is continuous in. Before moving to the proof of the above result, it is important to underst the assumption in theorem 4.1. This is necessary in order to guarantee that the quotient [, in turn, ] is well defined for all. However, even if this assumption is not satisfied for some, it is still possible to (6)

7 556 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 Fig. 11. System with relative degree 7. obtain a linear representation of the Poincaré map for all. Such linear transformation would be parametrized by another variable at, i.e.,. Proof: Let. Integrating the differential equation (2) gives As we will see next, based on this theorem, it is possible to reduce the problem of checking quadratic stability of Poincaré maps to the solution of a set of LMIs. The Poincaré map defined above is quadratically stable if there exists a symmetric matrix such that If then, i.e., Now, let. Let also be the switching time associated with. Then Using (7), the last equality can be written as (7) Success in finding satisfying (9) is then sufficient to prove global asymptotic stability of the limit cycle. A sufficient condition for the quadratic stability of a Poincaré map can easily be obtained by substituting (6) in (9) (9) (10) for some for all, with associated switching times. There are several alternatives to transform (10) into a set of LMIs. A simple sufficient condition is on (11) Since,, or, that is Therefore, it is also true that. Since, by assumption,, for some all is well defined for (for it is defined via continuation). Replacing above, we get for all. This result agrees with proposition 3.3. Via continuation, at is given by where. Using equality (7), can be written as. This means is exactly the Jacobian of the Poincaré map at. (8) for some for all, where on sts for for all nonzero. In the next section, using some illustrative examples, we will see that although this condition is more conservative than (10), it can prove global asymptotic stability of many important RFS. Other less conservative conditions are considered discussed in Section VI. These are based on the fact that is a map from to, that the set of points in with the same switching time is a convex subset of a linear manifold of dimension. Before moving into the examples, it is important to notice that condition (11) can be relaxed. Since is Hurwitz is a bounded input, there is a bounded set such that any trajectory will eventually enter stay there. This will lead to bounds on the difference between any two consecutive switching times. Let be bounds on the minimum maximum switching times of trajectories in that bounded invariant set. The expected switching times can, in general, be reduced to a smaller set. Condition (11) can then be relaxed to be satisfied on instead of on. See Section VII-A for details. V. EXAMPLES The following examples were processed in written by the authors. The latest version of this software is available at [13]. Before presenting the examples, it is important to underst these functions. Overall, the user provides an LTI system, together with, the hysteresis parameter. If the

8 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 557 RFS is proven globally asymptotically stable, the functions return a matrix that is guaranteed to satisfy (11) on, where, found as explained in Section VII-A, are bounds of the expected switching times. In more detail, after providing the software with an LTI system an hysteresis parameter, this confirms that certain necessary conditions are met. Then, it checks if there exists a unique locally stable symmetric unimodal limit cycle. This is done by first finding, the zeros of (4). A symmetric unimodal limit cycle exists if, for some, for all, is unique if this is true for only one. Before explaining the remainder of the functions, it is important to point out that, although the vectors are -dimensional, the solution generated by the Poincaré map is restricted to the -dimensional hyperplane (see Fig. 7). Therefore, the map is actually a map from to. Let be a map from to, where are the orthogonal complements to, i.e., matrices with a maximal number of column vectors forming an orthonormal set such that. An equivalent condition to (11) is then (12) for some symmetric matrix all switching times, where. in (11) can be obtained by letting. Equation (12) on forms an infinite set of LMIs. Computationally, to overcome this difficulty, we grid this set to obtain a finite subset of expected switching times. In other words, is found by solving a finite set of LMIs consisting of (12) on,. For a large enough, it can be shown that (12) is also satisfied for all. The idea here is to find bounds on the derivative of the minimum eigenvalue of over, to use these bounds to show that nothing can go wrong in the intervals, i.e., that (12) is also satisfied on each interval. Solving a set of LMIs allows us to find in (12). In the examples below, once is found, we confirm (12) is satisfied for all switching times by plotting the minimum eigenvalue of on, showing that this in indeed positive in that interval. Example 5.1: Consider the RFS on the left of Fig. 8. Since for this system any state-space realization of the LTI system in relay feedback results in, it is possible to consider the ideal relay, i.e.,. Although very simple, this system has never been proved globally stable. From the center of Fig. 8 it is easy to see the RFS has one unimodal symmetric limit cycle with period approximately equal to. We have analyzed this same RFS in [12]. There, we characterized a reasonably large region of stability around the limit cycle. Using the software described above, however, we were able to find a satisfying (12) for all switching times, showing, this way, that the RFS is actually globally asymptotically stable. The right side of Fig. 8 confirms the result. Fig. 12. Example of a set (in, both its image in are segments of lines). Example 5.2: Consider the RFS in Fig. 9. Let. As seen in Fig. 9, the RFS has one unimodal symmetric limit cycle with period approximately equal to. Again, a satisfying (12) for all switching times exists, which means the limit cycle is globally asymptotically stable. This is confirmed from the right side of Fig. 9. Example 5.3: Consider the 6th-order RFS in Fig. 10. In this case, sliding modes occur if. However, stability was proven for as low as Fig. 10 shows the result to. Note that, in the figure on the right, the function depicted is always positive although, due the bad resolution, it may seem otherwise. This is due to the fact that is the lowest value for which we can still prove global stability. It is interesting to notice that more than one limit cycle exists for. Thus, for this example, condition (11) is not conservative. Example 5.4: Consider the RFS in Fig. 11 consisting of an LTI system with relative degree 7 in feedback with an hysteresis, where. As seen in the center of Fig. 11, this RFS has a symmetric unimodal limit cycle with period, where. Note how the period of the limit cycle is much larger than the hysteresis parameter. Again, from the ride side of Fig. 11, we conclude that the limit cycle is globally asymptotically stable. VI. IMPROVEMENT OF STABILITY CONDITION As mentioned before, there are several alternatives to transform (10) into a set of LMIs. Here, we explore some of these alternatives to derive less conservative conditions than (11). The Poincaré map is a map from to, for each point in, there is at least one associated switching time. An interesting property of this map is that the set of points in with the same switching time forms a convex subset of a linear manifold of dimension. Let be that set, i.e., let be the set of points that have as a switching time, i.e., (see Fig. 12). In other words, a trajectory starting at satisfies both on,. Note that since is a multivalued map, a point in may belong to. In fact, in Example 3.1, there existed a more than one set point in that belonged to both. Condition (11) can then be improved to on (13) for some for all expected switching times.

9 558 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 Fig. 13. On the left: for ; on the right: for. The problem with condition (13) is that, in general, the sets are not easily characterized. An alternative is to consider the sets obtained from (8), given by To see the difference between, consider the example in Fig. 13 where the solution is plotted for two different initial conditions in. On the left of Fig. 13,. This means belongs to both,. The right side of Fig. 13 shows what would happen to if the trajectory had not switched at (dashed curve). In that case, it would have intersected again at. This means that although is a solution of (8), it is not a switching time since for. In other words, the switching time does not satisfy the inequality on. Although both satisfy (8), only is a valid switching time, i.e.,. Thus, belongs to,,, but it does not belong to. Since, condition (13) holds if there exist a such that Fig. 14. View of the cone in the plane. for all expected switching times. As seen in Fig. 14, on (14) satisfies a conic relation Fig. 15. System of relative order 7 with. for some matrix constructed. Let (Section VII-B explains how this matrix is It is important to notice that it is equivalent to say that some matrix satisfies on or that on. This has to do with the fact that quadratic forms are homogeneous. To see this, assume for all. Let where. Then, which is to say on. The converse follows since. Condition (14) is then equivalent to for some for all expected switching times. Using the -procedure, condition (14) is again equivalent to on on (15) for some, some scalar function, for all expected switching times. Note that, for each, (15) is an LMI. Example 6.1: Consider again the system with relative degree 7 analyzed in Example 5.4. For small values of there is no satisfying condition (11). Using condition (15), however, a a positive function satisfying (15) are known to exist for values of as small as Fig. 15 shows the result to. Again, the function depicted on the right in the figure is always positive although, due to bad resolution, it may seem otherwise. Note that the function on the left of the figure has three zeros. However, only one corresponds to a limit cycle. Although condition (11) was not able to prove global stability of the RFS for small values of, the less conservative condition (15) proved that the limit cycle is globally asymptotically stable for small values of. An interesting fact is that, for, there is more than one limit cycle. It is possible to improve condition (15) furthermore. This condition does not take advantage that a trajectory starting at

10 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 559 must satisfy on. This is captured by condition (13) but not by (15) since. Constraint on can be expressed as (16) for all. However, this last inequality would lead to an infinite dimensional set of LMIs. One way to transform the problem into a finite set of LMIs is to consider certain samples of time in. For instance, if, then we would have the following constraint on enter stay there (i.e., will not leave the set). Remember that, by definition, is given by Proposition 7.1: Consider the system,, where is Hurwitz,, is a row vector. Then, for any fixed Proof: At time is given by This, together with, satisfies a conic relation in which case (15) could be improved to on (17) Therefore for some scalar function. There is an infinite number of constraints that can be added to condition (17) in order to further reduce the level of conservatism. On one h, the more constraints, the better chances to find surface Lyapunov functions. On the other h, increasing the number of constraints will eventually make the problem computationally intractable. In spite of this, it is interesting to notice that many important RFS were proven globally stable with just condition (11) (the most conservative of all presented in this paper). We want to point out that the value of all these results lie in the fact that they work well. In fact, we have not been able to find RFS with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. This lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. VII. COMPUTATIONAL ISSUES In this section, we will talk about computational aspects related to finding in (11) (15). First, we show that since is Hurwitz is a bounded input, there is a bounded invariant set such that any trajectory will eventually enter. This will lead to bounds on the difference between any two consecutive switching times. This way, the search for in (11) (15) becomes restricted to. Then, we will talk about the cones used in Section VI. In particular, we describe how to construct. A. Bounds on Expected Switching Times For a fixed, condition (11) is an LMI with respect to, while (15) is an LMI with respect to. In this section, we want to show that it is sufficient that conditions (11) or (15) are satisfied in some carefully chosen interval, instead of requiring them to be satisfied for all expected switching times. In order to do so, one must guarantee there exists a such that the difference between any two consecutive switching times of a trajectory for is higher than but lower than. Before we find such bounds, we need to show there is a particular bounded set such that any trajectory will eventually which is equal to. We now focus our attention in finding an upper bound for. First, remember from the proof of theorem 4.1 that a trajectory starting at is given by. Then, the output is given by By definition of, at least in some interval, where. However, since we are assuming, Hurwitz, it is easy to see that cannot remain larger than for all. For any initial condition, as. Hence, since for sufficiently large time, is bounded (from the above proposition), an upper bound on on the expected switching times can be obtained. Proposition 7.2: Let be the smallest solution of (18) If are sufficiently large consecutive switching times then. Proof: Assume that after a sufficiently large time the trajectory is at. Without loss of generality, assume

11 560 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL Then will be positive in some interval. We are interested in finding an upper bound on the time it takes to switch. That is, we would like to find an upper bound of those such that, i.e., Using Proposition 7.1 with, we can get a bound on the left side of the inequality We are looking for values of such that. has two solutions However, only one is positive (the one with the sign) since for all either (if ) or (if ). To find we find a bound on the first derivative of for Therefore, must satisfy (18). Remember that if, will be positive at least in some interval. The next result shows that in the bounded invariant set characterized in proposition 7.1, cannot be made arbitrarily small. Basically, for sufficiently large time is bounded, a lower bound on the time it takes between two consecutive switches can be obtained. Proposition 7.3: Let,, define Also, let. If are sufficiently large consecutive switching times then. Proof: There are many ways to find bounds on. We will show two here:. Since they are found independently of each other, we are interested in the larger one. We start with. Assume again that after a sufficiently large time the trajectory is at. Without loss of generality, assume. This means that right before the switch (at ),, i.e.,. Therefore, after the switch at,. That is,. We also need bounds on the second derivative of for. From we get,. This means that So,. The worst case scenario is the case when [with ]. Therefore,. Again, we are looking for values of such that, i.e., the solution of. B. Construction of the Cones We now describe how to construct the cones introduced in Section VI. Let denote the boundary of, i.e.,. Remember that for each, the cone is defined by two hyperplanes in : one is the hyperplane parallel to containing the other is the hyperplane defined by the intersection of, containing the point (see Fig. 14). Let, respectively, be vectors in perpendicular to each hyperplane. Once these vectors are known, the cone can easily be characterized. This is composed of all the vectors such that. The symmetric matrix introduced in the definition of is just where. Remember that the cone is centered at note that after is chosen, must have the right direction in order to guarantee. We first find, the vector perpendicular to. Looking back at the definition of, is given by The derivation of is not as trivial as. We actually need to introduce a few extra variables. The first one is, the vector perpendicular to the set, given by Proposition 7.4: The hyperplane defined by the intersection of, containing the point is perpendicular to the vector So,. In order to find a lower bound on the switching time, we consider the worst case scenario, that is, we consider the case when. This implies that. Integrating once more knowing that, yields Proof: can be parameterize the following way

12 GONÇALVES et al.: GLOBAL STABILITY OF RELAY FEEDBACK SYSTEMS 561 The intersection of occurs at points in such that. Multiplying on the left by we have or We want to show that Using (19) we have (19) The characterization of is not complete yet. The orientation of must be carefully chosen to guarantee that the cone contains. Proposition 7.5: If then the cone contains. The proof, omitted here, is based on taking a point showing that. VIII. CONCLUSION This paper introduces an entirely new constructive global analysis methodology for PLS. This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to construct quadratic surface Lyapunov functions to show that maps between switching surfaces are contracting in some sense. These results are based on the discovery that maps induced by an LTI flow between two switching surfaces can be represented as linear transformations analytically parametrized by a scalar function of the state. Furthermore, level sets of this function are convex subsets of linear manifolds. This representation allows the search for quadratic Lyapunov functions on switching surfaces to be done by simply solving a set of LMIs. This methodology has proved very successful in analyzing a simple class of PLS known as RFS. We addressed the problem of global asymptotic stability of symmetric unimodal limit cycles of RFS with hysteresis. This is a hard problem since global analysis tools were practically nonexistent. However, with these new results, a large number of examples with a unique locally stable symmetric unimodal limit cycle was successfully globally analyzed. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. There are still many open problems following this work. It is currently under investigation how to apply this new methodology to globally analyze more general PLS, not only in terms of stability, but also robustness performance. Knowing that quadratic surface Lyapunov functions were so successful in analyzing RFS, we pose the question: can similar ideas be used to efficiently systematically globally analyze larger more complex classes of PLS? We suspect that the answer to this question is yes. We are currently working to support our conjectures. In fact, we have recently proved global asymptotic stability of equilibrium points of on/off systems [10] saturation systems [11]. We have also been able to check performance of on/off systems [9, Ch. 8]. Another important topic of research following this work is to find conditions that do not depend on the parameters of the Lyapunov functions but guarantees their existence. Such conditions should depend on the plant or on certain properties of a class of systems, should, obviously, be easier to check than the ones presented here. REFERENCES [1] S. H. Ardalan J. J. Paulos, An analysis of nonlinear behavior in delta sigma modulator, IEEE Trans. Circuits Syst., vol. CAS-6, pp , [2] K. J. Åström, Oscillations in systems with relay feedback, IMA Vol. Math. Appl.: Adapt. Control, Filtering, Signal Processing, vol. 74, pp. 1 25, [3] K. J. Åström K. Furuta, Swinging up a pendulum by energy control, in IFAC 13th World Congr., San Francisco, CA, [4] K. J. Åström T. Hagglund, Automatic tuning of simple regulators with specifications on phase amplitude margins, Automatica, vol. 20, pp , [5] D. P. Atherton, Nonlinear Control Engineering. New York: Van Nostr, [6] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System Control Theory. Philadelphia, PA: SIAM, [7] P. B. Brugarolas, V. Fromion, M. G. Safonov, Robust switching missile autopilot, in Amer. Control Conf., Philadelphia, PA, June [8] M. di Bernardo, K. Johansson, F. Vasca, Sliding orbits their bifurcations in relay feedback systems, in 38th Conf. Decision Control, Phoenix, AZ, Dec [9] J. M. Gonçalves, Constructive global analysis of hybrid systems, Ph.D. dissertation, Massachusetts Inst. of Technol., Cambridge, MA, Sept [10], Global stability analysis of on/off systems, in Conf. Decision Control, Sydney, Australia, Dec [11], Quadratic surface Lyapunov functions in global stability analysis of saturation systems, MIT, Technical report LIDS-P-2477, Aug [12] J. M. Gonçalves, A. Megretski, M. A. Dahleh, Semi-global analysis of relay feedback systems, in Proc. Conf. Decision Control, Tampa, FL, Dec [13] J. M. Gonçalves. [Online]. Available: [14] J. Guckenheimer P. Holmes, Nonlinear Oscliiations, Dynamical Systems, Bifurcations of Vector Fields. New York: Springer-Verlag, [15] A. Hassibi S. Boyd, Quadratic stabilization control of piecewise linear systems, in Amer. Control. Conf., Philadelphia, PA, June [16] K. H. Johansson, A. Rantzer, K. J. Åström, Fast switches in relay feedback systems, Automatica, vol. 35, no. 4, April [17] M. Johansson A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Automat. Contr., vol. 43, pp , April [18] A. Megretski, Global stability of oscillations induced by a relay feedback, in Preprints 9th IFAC World Congr., vol. E, San Francisco, CA, 1996, pp

13 562 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 4, APRIL 2001 [19] Y. I. Neimark, Methods of Pointwise Mappings in the Theory of Nonlinear Oscillations (in Russian). Moscow, Russia: Nauka, [20] S. Pettersson B. Lennartson, An LMI approach for stability analysis of nonlinear systems, in EEC, Brussels, Belgium, July [21] N. B. Pettit, The analysis of piecewise linear dynamical systems, Control Using Logic-Based Switching, vol. 222, pp , Feb [22] Y. Takahashi, M. J. Rabins, D. M. Ausler, Control Dynamic Systems. Reading, MA: Addison-Wesley, [23] C. Tomlin, J. Lygeros, S. Sastry, Aerodynamic envelope protection using hybrid control, in Amer. Control Conf., Philadelphia, PA, June [24] Ya. Z. Tsypkin, Relay Control Systems. Cambridge, U.K.: Cambridge Univ. Press, [25] S. Varigonda T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Contr., to be published. Jorge Gonçalves was born in He received the Licenciatura (5-year B.S.) degree from the University of Porto, Portugal, the M.S. Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, all in electrical engineering computer science, in 1993, 1995, 2000, respectively. From 2000 to 2001, he held a research position at the Massachusetts Institute of Technology. Since 2001, he has been with the California Institute of Technology, Pasadena, where is currently a Postdoctoral Scholar in the Control Dynamical Systems Department. His research interests include modeling, analysis, control of certain classes of hybrid systems robustness analysis of nonlinear systems. Dr. Gonçalves was the recipient of the Best Student Paper Award at the Automatic Control Conference, Chicago, IL, in June Alexre Megretski was born in He received the Ph.D. degree from Leningrad University, Russia, in He held research positions at Leningrad University, Mittag-Leffler Institute the Royal Institute of Technology, Sweden, the University of Newcastle, Australia. From 1993 to 1996, he was an Assistant Professor at Iowa State University, Ames. In 1996, he joined the Massachusetts Institute of Technology, Cambridge, where he is currently an Esther Harold E. Edgerton Associate Professor of Electrical Engineering. His research interests include automatic analysis design of complex dynamical systems, romized methods of nonconvex optimization, operator theory. Munther A. Dahleh was born in He received the B.S. degree from Texas A&M University, College Station, TX, the Ph.D. degree from Rice University, Houston, TX, all in electrical engineering, in , respectively. Since then, he has been with the Department of Electrical Engineering Computer Science, Massachusetts Institute of Technology, Cambridge, where he is now a Full Professor. He was a Visiting Professor at the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, for the Spring of He has held consulting positions with several companies in the U.S. abroad. He is the coauthor (with Ignacio Diaz-Bobillo) of the book Control of Uncertain Systems: A Linear Programming Approach (Englewood Cliffs, NJ: Prentice-Hall, 1995, the coauthor (with Nicola Elia) of the book Computational Methods for Controller Design (New York: Springer-Verlag, 1998). His interests include robust control identification, the development of computational methods for linear nonlinear controller design, applications of feedback control in several disciplines including material manufacturing modeling of biological systems. Dr. Dahleh was the recipient of the Ralph Budd Award in 1987 for the best thesis at Rice University, the George Axelby Outsting Paper Award (paper coauthored with J.B. Pearson) in 1987, an NSF Presidential Young Investigator Award in 1991, the Finmeccanica Career Development Chair in 1992, the Donald P. Eckman Award from the American Control Council in 1993, the Graduate Students Council Teaching Award in He was a Plenary Speaker at the 1994 American Control Conference. He is currently serving as an Associate Editor for IEEE TRANSACTIONS ON AUTOMATIC CONTROL.